Interfacial dynamics in rapid solidification processes

Physica A 280 (2000) 148–154www.elsevier.com/locate/physa

Interfacial dynamics in rapid solidi�cationprocesses

Massimo Conti∗, Umberto Marini Bettolo MarconiDipartimento di Matematica e Fisica, Universita’ di Camerino, Istituto Nazionale di Fisica della

Materia, I-62032 Camerino, Italy

Abstract

The broad interest in rapid solidi�cation processes originates from the variety of non-equilibriummicrostructures found in the solidi�ed materials; most familiar examples are dendrites, lamellareutectics, cellular and banded structures. In spite of the deep e�orts devoted to this subject bythe scienti�c community, the interfacial dynamics is far from being well understood. The aimof this article is to review some recent theoretical developments, which give a di�use pictureof the interfacial region. A new microscopic approach, based on the stochastic dynamics of alattice of Potts spins, is also discussed. c© 2000 Elsevier Science B.V. All rights reserved.

PACS: 64.60−i; 81.10.Aj; 64.70.DvKeywords: Rapid solidi�cation; Pattern formation

1. Introduction

Rapid solidi�cation phenomena play a major role in many �elds such as mate-rial engineering, crystal growth, chemistry and physics [1]. As the �nal properties ofthe solid material depend signi�cantly on the details of the growth process, the under-standing and prediction of the interfacial dynamics is of crucial importance. For puresubstances, the growth rate is controlled by the di�usion of the latent heat releasedat the solid–liquid interface; for alloy solidi�cation both heat and solute di�usion arethe limiting factors. The classical approach to describe the interfacial dynamics is for-mulated in terms of a moving boundary problem. The di�usion equation of heat (and

∗ Corresponding author. Fax: +39-0737-402524.E-mail addresses: [email protected] (M. Conti), [email protected] (U.M.B.Marconi)

0378-4371/00/$ - see front matter c© 2000 Elsevier Science B.V. All rights reserved.PII: S 0378 -4371(99)00631 -7

M. Conti, U.M.B. Marconi / Physica A 280 (2000) 148–154 149

eventually of solute) is complemented with boundary conditions at the moving front,re ecting two di�erent constraints: (i) conservation law for energy (and solute) at theinterface, and (ii) constitutive law which relates the chemical potential jump acrossthe interface to the growth rate. Point (ii) requires a separate modellization of theinterface kinetics. It should be mentioned that the contradiction between the sharpboundary conditions and the di�use nature of the interfacial region poses delicate prob-lems when the model is applied to the description of fast transient growth [2]. A morerecent method to investigate solidi�cation processes is based on the phase-�eld model(PFM) [3,4]. In this model a non-conserved order parameter characterizes the phase ofthe system at each point. A suitable free-energy (or entropy) functional is then con-structed, that depends on the order parameter as well as on the associated (conserved)�elds and their gradients. The functional derivatives of this functional with respect tosuch order parameter and �elds determine the evolution of the system towards equi-librium. Studies conducted on solidi�cation of both pure substances and binary alloys[5,6] pointed out that the PFM incorporates in a natural fashion the e�ects of inter-face curvature and non-equilibrium phenomena as the trapping of solute into the solidphase and the kinetic undercooling of the solid–liquid interface. Moreover, describingthe interface as a region of �nite thickness, gives a more natural and consistent pictureof the solidi�cation front.The existing approaches to the problem represent at di�erent degrees a coarse-grained

picture of the microscopic processes. In fact, thermal uctuations are not included ineither model. An alternative approach consists in starting from the level of descrip-tion of the Hamiltonian together with a microscopic dynamics of stochastic character,and to recover the observed macroscopic behavior at a macroscopic and mesoscopicscale without coarse graining the model [7,8]. This task avoids some of the tradi-tional di�culties such as the use of adhoc free-energy functional in the PFM and theneed to postulate sharp boundary conditions, in contradiction with the di�use nature ofthe interfacial region, in the free-boundary model. This microscopic statistical model(MSM) has been exploited in some recent studies on rapid solidi�cation of both puresubstances and binary alloys.Some of the most relevant results obtained starting from the PFM and MSM are

reviewed hereafter. Section 2 is devoted to solidi�cation of pure substances, andSection 3 to alloy solidi�cation. The concluding remarks follow in Section 4.

2. Solidi�cation of pure substances

Investigations on the solidi�cation of pure substances conducted through the phase-�eld model were at �rst directed to demonstrate that the model equations reducedasymptotically to the free-boundary formulation when the interface width tends tozero. The control parameter for the interface velocity v is the non-dimensional un-dercooling �, de�ned as � = cp(Tm − T∞)=L, where cp is the speci�c heat, T∞ thefar-�eld temperature and Tm the equilibrium melting temperature; L is the heat of

150 M. Conti, U.M.B. Marconi / Physica A 280 (2000) 148–154

fusion. One-dimensional numerical simulations [9] showed that the model properly de-scribed the di�usional regime for �¡ 1 (with v decaying as ˙ t−1=2) and the steadyregime for �¿1. Numerical studies of the growth of a single dendrite were conductedin two dimensions and attention was focussed on the selection mechanism of the den-drite tip operating state, which intrigued the scienti�c community for a long time. Itwas con�rmed that in free growth conditions in order to have a stable and steady tippropagation, anisotropy of the surface energy is required [10]: the dendrite tip radius� and the velocity vtip are related to the anisotropy strength � through the dependence�2 vtip ˙ �−7=4. When the growth occurs in a narrow channel, the intrinsic anisotropyinduced by the channel boundaries allows steady tip propagation even for isotropicsurface tension [11]. The phase-separation dynamics of an initially undercooled meltwas also studied numerically within the framework of the phase-�eld model. In thelater stage of the growth process, the evolution is dominated by the di�usion of theresidual heat in the system and leads to a scaling regime characteristic of a conserveddynamics, with characteristic power-law growth of the average domain size proportionalto t1=3 [12]. These numerical results were corroborated by the solution of a sphericalversion of the PFM which lead to the prediction of crossover from a t1=2 behavior toan asymptotic t1=3 growth law [13–15].The microscopic statistical model (MSM) is based on a lattice dynamics in which

a Potts spin variable describing the local phase is associated to each lattice point;the heat conduction mechanism is implemented adding to each site additional de-grees of freedom, the Creutz demons, which follow a conserved dynamics. The modelhas proved able to properly describe the di�usive (�¡ 1) and the kinetic (�¿ 1)regimes for the one-dimensional growth [7]. In two dimensions a morphological insta-bility results in a front pattern which characteristic length is in good agreement withthe predictions of the Mullins Sekerka analysis.

3. Alloy solidi�cation

The analysis of the microstructures formed in rapid solidi�cation of binary alloys,evidenced that the partition coe�cient k (i.e., the ratio cs=cl of the solute concentrationin the growing solid to that in the liquid at the interface) increases from the equilib-rium value ke towards unity at large growth rate. This phenomenon was termed “solutetrapping”. Aziz [16], starting from a di�usional analysis across a steadily moving inter-face, was able to determine a dynamic phase diagram; the dependence of the partitioncoe�cient on the growth velocity is given in the form

k(v) =ke + v=vd1 + v=vd

(1)

being ke the equilibrium value for a stationary interface, and vd a characteristic velocitydescribing the di�usional solute redistribution across the moving front; vd is generallyexpressed as vd = D=a, where D is the interface solute di�usivity and a is the widthof the interfacial layer.


Fig. 1. The oscillatory dynamics of the solidi�cation front.

The dynamic phase diagram is generally employed to provide the non-equilibriumboundary conditions needed to solve the free-boundary di�usional problem. However,this method hides a basic contradiction. The moving front is treated, under all respects,as in�nitely thin layer, whereas Eq. (1) originates from a di�use interface picture ofthe solidi�cation process (for a sharp interface a = 0, vd =∞ and no solute trappingshould occur). This ambiguity emerges in the description of time dependent processes:solute relaxation across the interface is not instantaneous but takes time of the order� ∼ a2=D = a=vd, and the free-boundary equations give an unrealistic picture of theprocess when the transient characteristic time is of the order of �. On the contrary, thephase-�eld model intrinsically accounts for �nite time e�ects in the interfacial dynam-ics. As an example [2], we refer to the oscillatory dynamics observed in some rapiddirectional solidi�cation experiments (i.e., solidi�cation driven by a thermal �eld whichmoves at constant velocity V0 with a thermal gradient G). Fig. 1 shows the results ofa phase-�eld simulation for a Ni–Cu alloy: the interface velocity is represented versustime, for V0 = 700 and G = 40 K (except for temperatures, dimensionless units areused, scaling lengths to the interface width a and times to a2=D. The process neverreaches a steady regime, and the interface velocity continuously oscillates around theaverage value V0. Here � is of the order of 10−5, and the fast transients shown inFig. 1 exhibit Fourier components comparable with 1=�; then we should observe abreakdown of the free-boundary picture. This suggestion is con�rmed by Fig. 2, wherethe cycle described by the partition coe�cient, as given by the phase-�eld simulation(solid dots), is compared with the predictions of Eq. (1) (solid line). We note thatthe partition coe�cient is not a uniquely de�ned function of the interface velocity, butshows an hysteretic behaviour and deviates from the free-boundary predictions during


Fig. 2. The partition coe�cient versus the interface velocity along the cycle described by the solidi�cationprocess.

a signi�cant portion of the cycle. In the MSM [8] for alloy solidi�cation the localphase is described through a discrete Potts state variable which follows a stochasticnon-conserved dynamics; the chemical species is associated with a spin variable chang-ing according to Monte Carlo spin-exchange dynamics in order to conserve the numberof particles of each species. These two variables allow to de�ne an Hamiltonian whichmodels a binary mixture undergoing a solid–liquid phase separation, driven by thecompetition between the larger entropy of the liquid and the lower energy of the solid.The mean �eld approximation of the model results in a lens-shaped equilibrium phasediagram, characteristic of ideal solutions. The o� equilibrium dynamics is character-ized by the di�erent frequencies with which one attempts to change the two variables,re ecting the two di�erent time scales of the solidi�cation process and of the solutedi�usion. The model has been solved in directional solidi�cation conditions. It wasobserved that the width of the interfacial layer is of the order of several lattice sites.Figs. 3 and 4 show the solute pro�le for two di�erent values of the rate of attempts fdfor the species variable. It can be observed that the concentration gap at the interfaceincreases with increasing fd. This is a manifestation of the trapping of solute into thesolid phase. At high di�usion rates the solute redistribution across the advancing frontis very e�ective and the partition coe�cient approaches the equilibrium value ke. Atlow values of fd the di�usional mechanism is too slow with respect to the growthrate and the solute has no time to be e�ectively rejected across the interface; as aconsequence the concentration gap decreases and the partition coe�cient approachesunity.


Fig. 3. The solute pro�le for directional solidi�cation. The rate of attempts for the species variable is fd=16per each Monte Carlo step; xI locates the interface position.

Fig. 4. The solute pro�le for directional solidi�cation. The rate of attempts for the species variable is fd=40per each Monte Carlo step; xI locates the interface position.

4. Conclusion

Di�use interface models give a realistic picture of rapid solidi�cation processes, ac-counting for non-equilibrium phenomena and �nite-time e�ects. Phase-�eld simulationsallowed to gain more insight into the interfacial dynamics underlying the formationof dendritic patterns and the late stage coarsening of solid islands. With respect to


the free-boundary approach alloy solidi�cation is more accurately described for fasttransient growth. The stochastic behavior intrinsic to the microscopic statistical modelproperly accounts for the uctuations; the model has proved able to predict the mor-phological instability of the solidi�cation front and non-equilibrium e�ects as solutetrapping.

References

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Interfacial dynamics in rapid solidification processes

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